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46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference 18 - 21 April 2005, Austin, Texas AIAA 2005-1870 AIAA 2005-1870 Bi-stable Cylindrical Space Frames H. Ye and S. Pellegrino University of Cambridge, Cambridge, CB2 1PZ, UK 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 18-21 April 2005 Austin, Texas For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344 Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Bi-stable Cylindrical Space Frames H. Ye∗ and S. Pellegrino† University of Cambridge, Cambridge, CB2 1PZ, UK This paper presents a novel kind of bi-stable structure: bi-stable space frames, based on a double-layer cylindrical architecture. An analytical method to carry out a preliminary assessment of the bi-stability of such space frames is introduced here and a series of bi-stable space frames found by this method are presented. Non-linear finite element simulations have been used to test one of the space frames and confirm the analytical predictions. Introduction Several bi-stable structures, which have two discrete stable configurations, have recently been found and the application of these structures to next-generation deployable and adaptive structures is currently being investigated at Cambridge. Many of the bi-stable structures that are known so far have the form of a metallic or composite cylindrical shells [1] [2] and assemblies of hinged bars and tape springs that are bi-stable are also known [3]. This paper will present a new kind of bi-stable structure –a bi-stable space frame– based on a doublelayer cylindrical architecture. A method to carry out a preliminary assessment of the bi-stability of this kind of structure will be introduced and explained. A series of space frames having bi-stable properties have been found. Several key issues concerning the design of this kind of bi-stable structure have been revealed. Based on the geometrical estimation method, a cylindrical space frame that is potentially bi-stable has been selected and a series of detailed geometrically non-linear finite element simulations of this structure, varying several design parameters, have been done to investigated its bi-stability. Finally, a couple of physical models built by rapid prototyping technology will be presented. we introduce the equivalent continuum modulus matrix for some simple examples. Consider the two-dimensional lattice structure shown in Figure 1(a). It is assumed that all squares forming this structure are identical, and have the equal side length L and axial stiffness AE. A Cartesian coordinate system has been defined for the structure, whose axes are parallel to the sides of the space frame. As in continuum mechanics, we can introduce the equivalent modulus matrix D to describe its in-plane structural properties. The relationship between the in-plane axial and shearing force per unit length and the corresponding deformation variables is given by Nx ²x Nxy = D ²xy (1) Ny ²y in which the matrix on the right of the equation is the equivalent modulus matrix: 1 0 0 AE 0 0 0 D= (2) L 0 0 1 Ny Ny Nxy Nxy Nxy 1 L Nxy 1 L Nx Nx Simple Geometrical Approach Structural properties of space frames A space frame is a structure system whose overall shape is two- or three-dimensional. It consists of linear elements that carry loads in a three dimensional way and these elements form a series of repeating and regular units [4]. For any space frame, the general structural properties can be usefully described in term of an equivalent continuum, which simulates the properties of basic repeating units in the space frame. Here ∗ Research Student. of Structural Engineering, Department of Engineering, Trumpington Street. Associate Fellow AIAA. † Professor Copyright ° c 2005 by H. YE and S. Pellegrino. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. 1 1 (a) (b) Fig. 1 Two dimensional repeating lattice based on (a) square and (b) right-angle triangle. Similarly, for the two dimensional space frame based on right-angle triangles shown in Figure 1(b), the equivalent modulus matrix D can also be obtained by superposing the contribution of the additional diagonal bar to the modulus matrix in Equation (2): 1 1 1 √ √ 1 + 2√ 2 2 2 2 2 AE 1 1 1 √ √ (3) D= 2√2 2 2 2 2 L 1 1 1 √ √ √ 1 + 2 2 2 2 2 2 1 of 6 American Institute of Aeronautics and Astronautics Paper 2005-1870 Modulus matrices for lattices having several other inplane forms have also also been derived [5]. KXY (Kx0,Kxy0) y0 My Mx (Kc0,0) x0 θ0 (Ky0,-Kxy0) Mx X (a) d KXY y0 y (Kx0,Kxy0) My (Kx,Kxy) Fig. 2 Three dimensional space frame with identical top and bottom layer Y For a three dimensional double-layer space frame as Figure 2 shows— in which the two layers are identical to the two-dimensional lattice considered previously (Figure 1(b)), but are now separated by a distance d, we can also develop an equivalent modulus matrix to describe its structural behavior under pure outof-plane bending and twisting. If plane sections are assumed to remain plane and bars connecting the top and bottom layers provides a shear-rigid connection between layers, the relationship between bending and twisting moment per unit length and the corresponding curvatures and twist in the structure can be related as: Mx κx Mxy = D∗ κxy (4) My κy where the modulus matrix D∗ can be given by modifying the modulus matrix of the corresponding lattice in Equation (2): d2 D 2 (5) It can be shown that Equation (5) is valid for any double layer space frame with equal top and bottom layers. Equations (2) and (5) show that the elastic structural properties of any double-layer space frame can simply be derived from their corresponding two dimensional architecture. Deformation of cylindrical space frame We now consider the main object of our investigation—a double-layer space frame with cylindrical shape, whose mid-plane lies on the surface of a cylinder. To investigate its bi-stability, it is necessary to know the strain energy of the structure in all deformed configurations. For simplicity, we initially assume that only a uniform and inextensional deformation of this space frame is allowed, and hence the mid-plane of the space frame has to remain on a cylindrical surface at all stages, as shown in Figure 3. A fixed global coordinate system X − Y has been set in Figure 3 while a local coordinate system x − y attached to the space frame will be allowed to rotate along with the structure itself. The initial configuration of the space frame is represented x θ 1/Kc (0,0) 2θ KX,KY 2θ0 (Kc,0) x0 X D∗ = KX,KY 2θ0 (0,0) 1/Kc0 Y (Ky,-Kxy) (Ky0,-Kxy0) (b) Fig. 3 Inextensional deformation of a cylindrical space frame: (a)Initial configuration (b) Deformed configuration) by the curvature of the initial cylinder κc0 and the initial rotating angle θ0 . As shown in Figure 3(b), the deformed configuration will be denoted by curvature of current cylinder κc and the rotated angle θ. The strain energy associated with any amount of bending and twisting of this structure (but note that the structure is not allowed to bend simultaneously in two different directions, because of the assumed inextensional deformation) is expressed by the equation ∆κx 1 0 0 1 U = [∆κx ∆κxy ∆κy ] D∗ 0 2 0 ∆κxy 2 0 0 1 ∆κx (6) where D∗ is the equivalent modulus matrix of the space frame. ∆κx ,∆κxy and ∆κy are the changes of curvature of the structure in the local coordinate system. ∆κx ∆κxy ∆κy = κx − κx0 = κxy − κxy0 = κx − κy 0 (7) (8) (9) These curvatures can be obtained by Mohr’s circle (Figure 3): κc0 (1 + cos(2θ0 )) 2 κc0 = − sin(2θ0 ) 2 κc0 = (1 − cos(2θ0 )) 2 κx0 = κxy0 κy 0 (10) (11) (12) and κx κxy κy κc (1 + cos 2(θ0 + θ)) 2 κc = − sin 2(θ0 + θ) 2 κc = (1 − cos 2(θ0 + θ)) 2 = (13) (14) (15) For any given cylindrical space frame, with specified initial configuration κc0 and θ0 , we can consider 2 of 6 American Institute of Aeronautics and Astronautics Paper 2005-1870 all possible inextensionally deformed configurations by varying the curvature of the cylinder κc and the twisting angle θ. Therefore, the strain energy of all configurations can be calculated and their stability can be examined from the contour plot of the energy. 5 180 13 Second 9 11 5 13 7 θ (deg) 0.2 3 7 5 60 11 13 11 7 5 5 7 1 2 1 3 4 5 6 3 1 0.2 8 7 13 7 9 . 3 0 9 11 Initial stable configuration 9 9 20 5 25 13 40 1 13 9 11 9 5 3 1 3 7 5 9 10 κ (m−1) c Fig. 4 Contour plot of strain energy of a space frame with lattice corresponding to Figure 1(a). κc0 = 8.3 m−1 θ0 = 45◦ L = 8 mm d = 6 mm A = 0.196 mm2 E = 2.5 GPa 180 0.2 6 11 160 2 4 13 6 9 8 9 140 40 30 11 80 30 . 25 13 9 8 θ (deg) 25 35 9 100 9 60 35 40 11 1 2 3 98 . 2 6 0 5 13 6 4 11 13 20 4 40 Initial stable configuration 2 3 9 8 5 6 0.2 8 7 0 1113 9 6 8 4 2 9 10 −1 κc (m ) Fig. 5 Contour plot of strain energy of a space frame with lattice corresponding to Figure 1(b) of diagonal bars (as shown in Figure 1) may have crucial effects on determining the second configuration. The energy analysis method has been applied on a series of space frames to investigate the effects of the axial stiffness of diagonal bars. The results are summarized in Table 2. Here the axial stiffness of diagonal bars is denoted as AE1 while those of bars on the sides of squares are denoted as AE2 . The results clearly show that the second configuration is determined by the ratio of axial stiffness. With the increasing of the ratio, the curvature of second configuration increases up to the same value as the initial configuration. The rotating angle θ, however, remains the same. Table 2 Design rules for bi-stable space frames The fact that the above two space frames with different in-plane architectures have two varied second stable configurations indicates that the axial stiffness 11 Second stable configuration 120 11 the strain energy analysis method mentioned above has been performed to investigate the bi-stability. The contour plots of the strain energy of each space frame under inextensional deformation are shown in Figure 4 and Figure 5 respectively. It is clearly shown that there are two local minimum energy points in each diagram, which correspond to two stable configurations of each structure. Figure 4 shows the energy of a double-layer space frame with the same in-plane architecture as shown in Figure 1(a). One local minimum strain energy point is at κc0 = 8.3 m−1 and θ0 = 0, corresponding to the initial stress-free configuration. The other local minimum energy point is at κc1 = 8.3 m−1 and θ = 90◦ . By referring to Figure 3, it is found that the mid-plane of the second configuration lies on the same cylinder as the initial one, but rotated by 90◦ in the X − Y plane. Figure 5 shows the energy of a double-layer space frame with the same in-plane architecture in Figure 1(b). One local minimum strain energy point in this figure is also at initial configuration κc0 = 8.3 m−1 and θ0 = 0. The second local minimum energy point is at κc1 = 3.7 m−1 and θ = 90◦ . Here, the mid-plane of the second stable configuration lies on a cylinder different to the initial state. It is also observed that in Figure 4, both local minimum energy points have the same energy value. This space frame is therefore a symmetrically bi-stable structure, meaning a structure that has two stable states with the same energy level. In Figure 5, the two stable configurations have two different energy levels. This space frame is therefore a asymmetrically bi-stable structure [6]. 8 13 11 2 6 9 8 13 4 13 Initial curvature of mid-plane Initial rotating angle Side length of square Depth of space frame Cross section area (all bars) Young’s modulus of material . 1 13 9 7 5 3 80 25 stable configuration 11 7 9 We choose two example cases here to demonstrate this method of energy analysis. Two cylindrical double-layer space frames with the in-plane two dimensional architectures shown in Figure 1(a) and (b) have been selected and their structural properties are set as shown in Table 1. For these two space frames, Structural properties of space frames 11 13 7 11 120 5 7 911 13 9 9 Two examples of bi-stable space frame 1 3 5 140 100 Table 1 0.2 1 3 7 160 Effects of axial stiffness AE2 /AE1 0.5 1 2 3 5 10 ∞ κc0 (m−1 ) θ0 (deg) 8.3 45 8.3 45 8.3 45 8.3 45 8.3 45 8.3 45 8.3 45 κc1 (m−1 ) θ(deg) 2.7 90 3.7 90 4.7 90 5.5 90 6.3 90 7.2 90 8.3 90 Further research on additional space frame with other in-plane architectures has also been carried out. 3 of 6 American Institute of Aeronautics and Astronautics Paper 2005-1870 The parameters of the models are set to the same as in Table 1. The analysis results show that for the space frames with the first two in-plane architectures (Figures 6(a) and (b)), the second stable configuration remains the same at κc1 = 8.3 m−1 and θ = 90◦ . But for a model having the architecture shown in Figure 6(c), κc1 = 5.4 m−1 and θ = 79◦ . For the model with architecture shown in Figure 6(d), κc1 = 5.2 m−1 and θ = 94◦ . Combining the above results, it is demonstrated that the second configuration of this kind of bi-stable spaces frame is determined by the axial stiffness and the orientation of the diagonal bars. In other words, it is the diagonal bars that decide the shape of the second configuration. the parameters specified as the same as those in the simple geometrical calculation presented in Table 1. Ny Ny Nxy 1 Fig. 7 frame Nxy Nxy Nxy 1 2L L Nx L 2L Simulation techniques 1 1 (a) (b) The coordinates of the nodes in the structure for each of the previously obtained configurations have been calculated. In the FE analysis, the structure has been switched from the initial configuration to the second one by applying temporary displacement constraints on 5 nodes in the structure. After the second configuration had been approached sufficiently closely, all temporary constraints are removed, and only six rigid body constraints are left, in order to test the stability of the structure in this new configuration. All the analysis is carried out as nonlinear geometrical simulations in ABAQUS. Ny Ny Nxy Nxy Nxy Nxy 1 2L L L 2L Single units of a three-dimensional space 1 1 (c) (d) Nx Fig. 6 Analysis results and discussion Finite Element Modelling Based on a series of assumptions, the above geometrical approach gives a preliminary assessment of the bi-stability of cylindrical space frames. A more realistic finite element model has been established here to confirm and test the bi-stability of space frame. The analysis is carried out with the ABAQUS [7] finite element package. FE model establishment We choose a space frame with an in-plane square lattice in Figure 1(a) as an example to run the FE analysis. The initial geometrical assessment has been carried out in the previous section and the contour plot of strain energy of the space frame is shown in Figure 4. In the FE model, the first thing to do is to determine the three dimensional architecture of the space frame because so far only the pattern in the top layer and bottom layer has been considered. In order to remove the internal mechanism in the structure, some triangular forms have been introduced to connect the top layer and bottom layer. A single three-dimensional lattice has been presented in Figure 7 to represent one cell unit of the space frame. A 12x12 double-layer cylindrical space frame model was then established with all Initially, truss element T3D2 was used in the analysis. The ABAQUS simulation shows that the space frame does gradually move to the second configuration under the displacement constraints of 5 nodes. Figure 8 shows the top view of the initial configuration of the space frame and Figure 9 shows the top view of the final configuration after the simulation is completed, corresponding to the second stable state. It is noted that, in the initial configuration, the axis of the cylinder on which the mid-plane of the space frame lies, is direction 2 , while in the second configuration it has rotated by 90◦ to direction 1. The diagram of the strain energy variation during the FE analysis is shown in Figure 10. The energy increases from initially zero energy to the intermediate high energy level and goes down to almost zero level at the end of STEP 12. After that, only 6 rigid body constraints are retained on the space frame and the energy level decreases to zero at the end of STEP 13. The ABAQUS results exactly match the results from the simple geometrical estimation. Beam element B31 has also been used to model this space frame. The analysis procedure is the same as before. Figure 11 presents the energy change during 4 of 6 American Institute of Aeronautics and Astronautics Paper 2005-1870 the simulation. Because beam elements can endure bending deformation in addition to axial deformation, there is more energy stored in the structure when the displacement constraints have been applied. Therefore it can be observed in Figure 11 that the energy level at the second configuration is around half of the peak value. Also due to the beam elements, it is found that the ratio of diameter of bars to their length is crucial for the simulation to converge. Increasing this ratio may result in the analysis not converging. It is not reasonable to predict the stability of the second configuration in this case. −3 9 2 x 10 8 7 1 Fig. 8 Strain Energy (Nm) 3 Initial stable configuration. 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Step Fig. 11 ments. Strain energy variation using beam ele- Physical models Research into the constructing of physical models of these bi-stable space frames has also been carried out in Cambridge. To achieve monolithic structures, rapid prototyping technology has been applied to build these models. Two sample models based on the information gained from ABAQUS analysis have been selected to be built. The structural parameters and material properties of model 1 are shown in Table 3. These parameters are selected to ensure the ABAQUS simulation can converge when both truss or beam elements are used. The analysis shows that during the change of the configuration, the maximum von Mises equivalent stress in the entire model is 20 MPa for truss elements and 11 MPa for beam elements. Both cases are lower than the maximum allowed stress 69 MPa. 2 3 1 Fig. 9 Second stable configurations. −3 8 x 10 7 Strain Energy(Nm) 6 5 Table 3 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Step Fig. 10 ments. Parameters of model 1 Initial curvature of mid-plane Initial rotating angle Side length of square Depth of space frame Radius of cross section of bars Name of mateial Tensile modulus of material Tensile strength of material Elongation @ break Strain energy variation using truss ele- κc0 = 8.3 m−1 θ0 = 45◦ L = 8 mm d = 6 mm r = 0.25 mm SI50 (Resin) E = 2.48 − 2.69 GPa 48 50 MPa 5.3 − 15.0% Table 4 presents structural parameters and the material properties of model 2. During the simulation, 5 of 6 American Institute of Aeronautics and Astronautics Paper 2005-1870 Table 4 Parameters of model 2 Initial curvature of mid-plane Initial rotating angle Side length of square Depth of space frame Radius of cross section of bars name of mateial Tensile modulus of material Tensile strength of material Elongation @ break κc0 = 8.3 m−1 θ0 = 45◦ L = 10 mm d = 10 mm r = 0.5 mm Polyamide PA2200 (Nylon 12) E = 1.55 − 1.85 Gpa 42 − 48 Mpa 15 − 25% the maximum von Mises equivalent stress in the model is 42 MPa for truss elements and 48 MPa for beam elements. These also below the maximum allowable stress. Conclusions and Future work This paper has described double-layer cylindrical space frames which have two switchable stable configurations. The bi-stability of the structure was first analyzed by using a simple geometrical analysis. After that, a more refined Finite Element analysis was conducted, based on the initial estimation. This analysis has shown that the second configuration is indeed stable. Additionally, research on constructing physical models of this kind of bistable space frames has also been presented. Future work will involve investigation on space frames with other architectures. Further research must also be carried on to determine more suitable materials and manufacturing technologies for the construction of the bi-stable space frames. Acknowledgments Financial support from Cambridge-MIT Institute (CMI), Churchill College and Cambridge University Engineering Department is gratefully acknowledged. 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ABAQUS Theory and Standard User’s Manual, Version 6.4, Pawtucket, RI, USA. 6 of 6 American Institute of Aeronautics and Astronautics Paper 2005-1870